4.4E: Exercises - Quadratic Functions and Applications
- Page ID
- 147269
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
For each of the following exercises, determine if the parabola opens up or down.
- \(f(x)=-2 x^{2}-6 x-7\)
- \(f(x)=6 x^{2}+2 x+3\)
- \(f(x)=4 x^{2}+x-4\)
- \(f(x)=-9 x^{2}-24 x-16\)
- \(f(x)=-3 x^{2}+5 x-1\)
- \(f(x)=2 x^{2}-4 x+5\)
- \(f(x)=x^{2}+3 x-4\)
- \(f(x)=-4 x^{2}-12 x-9\)
- Answer
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1. down
3. up
5. down
7. up
In the following functions, find each of the following. Round answers to the nearest tenth where necessary.
- The vertex of its graph
- The equation of the axis of symmetry
- The x-intercept(s)
- The y-intercept
- The domain
- The range
9. \(f(x)=x^{2}+8 x-1\)
10. \(f(x)=x^{2}+10 x+25\)
11. \(f(x)=-3x^{2}+2 x+9\)
12. \(f(x)=-2 x^{2}-12 x-3\)
- Answer
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9.
- Vertex: \((-4,-17)\)
- Axis of symmetry: \(x=-4\)
- x-intercepts: \((-8.1,0), (0.1,0)\)
- y-intercept: \((0,-1)\)
- Domain: \((-\infty,\infty)\)
- Range: \([-17, \infty)\)
11.
- Vertex: \((0.3, 9.3)\)
- Axis of symmetry: \(x=0.3\)
- x-intercepts: \((-1.4,0), (2.1,0)\)
- y-intercept: \((0,9)\)
- Domain: \((-\infty,\infty)\)
- Range: \((-\infty,9.3]\)
In the following exercises, find the maximum or minimum value of each function.
13. \(f(x)=2 x^{2}+x-1\)
14. \(y=-4 x^{2}+12 x-5\)
15. \(y=x^{2}-6 x+15\)
16. \(y=-x^{2}+4 x-5\)
17. \(y=-9 x^{2}+16\)
18. \(y=4 x^{2}-49\)
- Answer
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13. The minimum value is \(−\frac{9}{8}\) when \(x=−\frac{1}{4}\).
15. The maximum value is \(6\) when \(x=3\).
17. The maximum value is \(16\) when \(x=0\).
In the following exercises, solve. Round answers to the nearest tenth.
19. A computer store owner estimates that by charging \(x\) dollars each for a certain computer, he can sell \(40 − x\) computers each week. The quadratic function \(R(x)=-x^{2}+40 x\) is used to find the revenue, \(R\), received when the selling price of a computer is \(x\), Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.
20. A cell phone company estimates that by charging \(x\) dollars each for a certain cell phone, they can sell \(8 − x\) cell phones per day. Use the quadratic function \(R(x)=-x^{2}+8 x\) to find the revenue received per day when the selling price of a cell phone is \(x\). Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.
21. A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the “walls.” He wants to maximize the area using \(80\) feet of fencing. The quadratic function \(A(x)=x(80-2 x)\) gives the area of the patio, where \(x\) is the width of one side. Find the maximum area of the patio.
22. A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them \(300\) feet of fencing to use to enclose part of their backyard. Use the quadratic function \(A(x)=x(300-2 x)\) to determine the maximum area of the fenced in yard.
23. A ball is thrown vertically upward from the ground with an initial velocity of \(109\) ft/sec. Use the quadratic function \(h(t)=-16 t^{2}+109 t+0\) to find how long it will take for the ball to reach its maximum height, and then find the maximum height.
- Answer
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19. \(20\) computers will give the maximum of $\(400\) in receipts.
21. The maximum area of the patio is \(800\) feet.
23. In \(3.4\) seconds the ball will reach its maximum height of \(185.6\) feet.